[41] Richard K. Guy: “Unsolved Problems in Number Theory”, Springer Verlag,New York, 1994, see pages 158 to 166 which include more than three pages of a bibliography on this subject that “shows only a fraction of what has been written”.

[42] Richard J. Gillings: “Mathematics in the Time of the Pharaohs”, first published 1972, edition consulted by Dover Publications, New York, 1982, pages 44 and 47.

When trying to reconstruct the role of mathematics in ancient Egypt we are not reduced to mere speculations.We have ancient testimony and other indirect evidence for the ancient Egyptians’ keen interest in mathematics.

Although the Classical Greeks, like many other provincial- minded people before and after them, liked to boast about the superiority of their own civilization, some of them explicitly acknowledged the Egyptians’ gold standard reputation for mathematical skills.

Herodotus, Proclus in his Commentary on Euclid[31], as well as Aristotle and Democritus all credited their neighbors in the Nile valley with the invention of geometry and with excellence in its practice[32].Also, Plato relates that the Egyptians taught arithmetic to their children with lessons based on enjoyment and games[33], so at least some of them must have kept that zest for the exploration of numbers as they grew older.

The same delight in mathematics for its own sake is well documented in ancient Mesopotamian tablets, together with the early number mysticism that was once as inseparable from mathematics as astrology from astronomy.

Similarly, even the minuscule sampling of mathematical documents that remain from pharaonic Egypt contains several instances of what looks like recreational mathematics -- “think- of- a- number” type problems presented with no practical purpose except for the pleasure of puzzling.Even van der Waerden acknowledged this leisure- requiring pursuit:

“The aha- calculations [algebraic riddles involving linear equations with one unknown] are not based on practical problems; they bear witness to the purely theoretical interests of the Egyptian computers.They have obviously been set up by people who enjoyed pure calculations and who wanted to drill their pupils in really hard problems.Like every art, arithmetic strives for its highest development.”[34]

Although some of the mathematical discoveries made by those pupils and their teachers may have been lost to others before Seshat's "Houses of Life" began to serve as formal systems of transmission, at least some of their insights must have slowly accumulated into a growing body of preserved knowledge about numbersand about the systems of arithmetic to manipulate these.

Already by Old Kingdom times, Egyptian number researchers had developed working systems of multiplication and divisionthat were so ingenious, so simple, and so efficient that the Greeks and many others continued to use them for many centuries after the last pharaoh was buried.

The Egyptians multiplied mostly by repeated doubling and adding, sometimes also by using ten as an intermediate multiplier when this was more convenient. This way, they avoided the need to learn multiplication tables[35].Their method has often been disparaged as awkward because it is different from the way we do our longhand multiplication -- it is much closer to the way our binary computers work.

Gillings lists in his book on “Mathematics in the Time of the Pharaohs” several disparaging comments from other authors on how unwieldy they find this Egyptian method but then asks:

“If Egyptian multiplication was so clumsy and difficult, how did it come about that these same techniques were still used in Coptic times, in Greek times, and even up to the Byzantine period, a thousand or more years later?No nation, over a period of more than a millennium [plus the at least three millennia since their invention], was able to improve on the Egyptian notation and methods.”[36]

A group of other authors relates an even longer survival of this early and easy- to- learn method:

“... as late as the Middle Ages doubling and halving were encountered as separate operations.In fact, under the heading of duplation (or duplication) and mediation they can be found as separate chapters in early American textbooks. (...)

There is an exact parallel to the Egyptian multiplication in the Russian peasant method of multiplication, said to be still in use today in some parts of Russia.”[37]

The Egyptians’ system of division into sums of unit fractions bears similar witness to its authors’ ingenuity and thoroughness.It, too, was so simple and worked so well that the mathematically oh so much more advanced Greeks, these darling inventors of science and of the good and the beautiful and the true, continued to follow that Egyptian system[38] rather than coming up with their own or using the more cumbersome[39] Babylonian sexagesimal method.

In that Babylonian method, divisions were easy only by divisors of the number base 60. All others were more difficult due to the great size of that base which required a multiplication table with theoretically 1740 entries (59 x 59 / 2) to be learned or consulted. This was such a burden that in practice, the Babylonians usually made do with smaller tables that left out the most easily remembered values.

But while the original Egyptian system was quite user- friendly, the copy the Greeks made of it represented a step backwards and worked at best as a buggy beta version: like the Hebrews, they replaced the simple and clear Egyptian number signs with letters from their alphabet.This double use of the letters was so confusing that Thomas Crump, a mathematician turned anthropologist, says the Greeks cut themselves off from the mainstream of numerical development in Babylon and thus blocked the development of mathematics in the Western world for more than a thousand years[40].

In our time, Egyptian fractions are still a subject of great interest to mathematicians and continue to provide these with many new and challenging questions, quite a few of which are still unsolved[41].

To divide a number, the Rhind Papyrus instructs its readers to add the divisor to itself until they reach that number.The number of times they have to add it is the result.The principle is the same as for the division we learned in school: first we guess how often the divisor goes into the number to be divided, then we multiply by that guess and deduct the result to repeat the process with the remainder.

When the Egyptians encountered a remainder, they expressed it as the sum of up to four different unit fractions in decreasing order that they looked up in tables just as we used to look up logarithms before we switched to electronic calculators.The Rhind Papyrus includes such a table for all odd divisors of two from 3 to 101, with none of the fractions smaller than 1/1000.

According to Gillings, this is “the most extensive and complete of all the arithmetical tables to be found in the Egyptian papyri that have come down to us”.That table is so useful in dealing with fractions that he says:

“However more sophisticated and advanced the mathematics of the Greeks, Romans, Arabs, and Byzantines may have been, not one of these nations over this long period of time had been able to devise a more efficient technique for dealing with the simple common fraction p / q.(...)

How was it possible for [the Egyptians] (...) to calculate unit fractional equivalents of 2/5, 2/7, 2/9 ..., 2/101 without a single arithmetical error?And how did it come about that, of all the many thousands of possible answers to these decompositions, those recorded by the scribe were in almost every case the simplest and best possible, by his own prescribed standards?”[42]

Many Egyptologists and mathematicians have puzzled for many years how the early Egyptians could have achieved this remarkable feat[43].

Here again, Gillings offers a collection of comments from prior authors, this time with widely diverging opinions about this system of fractions that range from “very beautiful” to “as useless as it was ambiguous” and “a monument to the lack of scientific attitude of mind”.Then he lists the five partly logical and partly aesthetic principles that seem to have guided the scribes’ decompositions into unit fractions, and he describes a computer analysis of all the 22,295 possible expressions of the Rhind table divisors.

In comparing the Rhind scribe’s solutions one by one to those offered by the computer, Gillings keeps complimenting the ancient sage for “an amazingly successful search”, for his unerring skill in finding the best solutions that makes us “wonder just how he did it”, and he adds:

“... one can only remain lost in hopeless admiration of the ancient Egyptian scribe, who could, with the meager arithmetic tools at his disposal, so unerringly locate this value. (...) Summing up, all the decompositions of the [Rhind] table, from one point of view or another, are the very simplest of all the decompositions possible.”[44]

With this level of mathematical skill and dedication to finding the best solutions, it is clear that the ancient Egyptians must have been at least as curious about numbers as any of their modern counterparts, or probably even more because they thought numbers had magic and religious and thus vitally important meanings.

Some mainstream scholars may object to this conclusion because this postulated importance of numbers does not appear in the mathematical writings from ancient Egypt. However, their argument collapses once they admit just how few of these writings survive.